Benne de Weger, the Netherlands|
"The title may be translated as The
Counting Devil, or maybe The Number Devil, and it has a subtitle that
translates to 'a pillowbook for everyone
who is afraid of math'. Enzensberger is a respected German novelist (not at
mathematician) who wrote this book to show to children that serious math is
fun and not difficult. It is about a schoolboy who in his dreams meets the
counting devil. Together they explore a variety of mathematical things (a
lot of them from
number theory, such as prime numbers, Fibonacci numbers and the golden
section, permutations, etc.), and it is written in such a way that young
children (such as my 8
year old son) will understand and enjoy it."
I finally read this book (Feb 2001)! It really was very nice, and none of the reviews I had read previously prepared me for the beautiful color illustrations throughout. My personal favorite among the boy's dreams is the one in which he learns about both the "Golden Mean" and the Euler characteristic. Though both of these things can be presented in a much more sophisticated way, it is wonderful how they can be delivered so casually, as if you were just saying "Hey, look, if you get out your calculator or draw a few figures on paper, you can notice some cool things!" I was a bit troubled by the fact that the author renames things, making it difficult for the reader of this book to continue their reading elsewhere. (For instance, prime numbers are `prima donnas', irrational numbers are `unreasonable' and Felix Klein is `translated' into Dr. Happy Little.) However, when I got to the end of the book I found that there is an index which also retranslates these terms back into their more familiar forms and was greatly relieved.
This book is the best - I am reading it to my 9 year old daughter - one chapter a day and she loves it. This is the BEST book ever on math and I have read many of them. Anything that gets a kid excited about math works for me. My daughter could not wait to get her calculator and find the square root of 2. I wish the author would write another book - it is wonderful. I am going to purchase the software - has anyone used it that would like to tell me about it?
Oh, hey! This is the first time I've heard about the software myself. Apparently there is a computer game based on the book. I'm tempted to buy a copy right now, just based on the fact that the book was so great. However, I'd be disappointed if it turns out to be nothing more than the book, read aloud and reproduced page by page. In particular, I'm hoping that it is very interactive and contains features that let curious kids go beyond what they may have already seen in the book. Can anyone write in to tell me (and Sue, above) how it is before we put our money down on it?
"As the Math Club facilitator for a Jr. Hi group, I consider this book an
incredible treasure. Math contests require this knowledge and here is a
fun way to present
it. I am always looking for new materials."
"After reading The Number Devil, I
recommended it to the 6th grade
students in my math class, and THEY
LOVE IT! The find it humorous, and
imaginative, as they discover the world
beyond the math book."
" I learned as much as my children. It
was one of the first books I read that
helped me realize how simple and
beautiful Math is!"
"It was a pleasure reading [this] book. It helps to capture the magic of
numbers. My mother raised me with a love of numbers and math and its
mysteries and complexities from an early age. Few books are written that
convey the wonder of numbers. [This one] does."
I love this book!It's very fun to read and it puts mathematics in a fun position.
Mary Jo Amschler|
I have questions! Please help!
I teach three classes of 9th grade students with a history of below average math skills. I just read The Number Devil this summer and am in the process of asking my school to purchase a class set so I can begin to integrate the chapters into my curriculum. I do love the book's (ok, really it's the author's) imagination!
My question is about irrational numbers ( "unreasonable numbers" as they are called in the book). In the index of the book, "unreasonable numbers" are translated as irrational. I think there is a problem here! Though I found the math, explanations and diagrams mathematically sound, I did not find the definition of irrational numbers sound. In chapter 4, didn't the author define the fractions 7/11 and 6/7 as irrational, when they are really rational? In chapter 10, are not all the quotients of the neighboring terms of the Fibonnaci Series rational? He defined them as irrational.
Please help! I fear I'm missing something, here!
Dear Mary Jo:
No, I don't think he has actually said anything wrong. Perhaps he has not been careful enough to make his point clear, but if you read closely you can see that he never makes the mistake of calling a rational number "unreasonable".
On page 74 where Robert divides 7 by 11 and 6 by 7, he gets two numbers which (obviously) are rational. He notes that there is a repeating pattern to their decimal expansion (a repeating 36 in the former case and a repeating 857142 in the latter). This is when the Number Devil first introduces the idea of irrational numbers by saying that there are others that are worse than these (rational) ones:
|(quoted from The Number Devil (Der Zahlenteufel))|
And there are many others that behave even worse, that go off the deep end after their zero. They are called the unreasonable numbers, and the reason they're called that is that they refuse to play by the rules.
He goes on to suggest that Robert take the square root ("rutabaga") of 2 and notes that he does not see any repeats in the decimal expansion. Now, one could argue that this is not a valid proof that the square root of 2 is irrational, since it could possibly just have a repeating pattern that is not apparent in the display of Robert's calculator (just as it is possible that the apparent pattern of 7/11 breaks down somewhere outside of the display on his calculator). But, he does not say that 7/11 or 6/7 are unreasonable...he is trying to draw a contrast between those rational numbers and the irrational square root of 2.
In the case of the ratios of Fibonacci terms, Robert is looking at a list and notes that 89/55 has a repeating 18 in its decimal expansion, but describes 21/13 as "looking as unreasonable as they come". Of course, 21/13 is not irrational, and perhaps Robert should know that, too. However, he doesn't actually say it is unreasonable, only that the decimal expansion looks unreasonable. In other words, this is an example, such as the type I warned about in the previous paragraph, which might look irrational on a calculator since the repeating pattern is not apparent. The author further confuses things by pointing out that the limit as n goes to infinity of the ratio of the n and n+1 term in the sequence actually is an irrational (unreasonable) number. I will agree that this may muddle things to the point that someone who is naive about number theory would become confused. However, looking at it myself now, I cannot see that he ever says anything that is actually wrong.
At least, that's the way it looks to me. I hope that helps!
(Note that this book is reviewed in the AMS Notices.)
I am a 9th grader who just recently had to write a paper about fibonacci numbers. I remembered reading this book a few years ago, and remembered the fibonacci numbers in this book. While it didn't give me much about fibonacci besides a few interesting little bits about his numbers, I did recall what a great read this was when i was in 6th grade. To this day i remember little things from this book, about the golden ratio and the many ways it appears in nature, the prime numbers, etc. I am astounded by how interesting this book made everything sound. Looking back on it now, I think this book was one of the main reasons i took such a liking to math. I would think to myself, "Now this is what math can be like, once you get past all the 'eight times seven equals fifty-four...no, wait...fifty-six' type of stuff. When reading such books as "The DaVinci Code", somehow it makes me feel really powerful to know that in a star with a pentagon around it, the ratio of the pentagon's sides to the star's legs is the golden ratio. This book is a very clever book that shows young kids a more exciting side to math.
My 5 and 7 year old kids love THe Number Devil, they would beg me to read it some more to them. Although some of the concepts may be a bit too deep for my 5 year old daughter but the way the concepts are presented in a playful way where kids can relate to is excellent.
However, I have to say that the number devil is a little on the sarcastic side which, if he was a real person, would turn off a mathophobic middle-schooler.
Overall, my husband, who's not fond of math, thought it was a fun math reading he thought he missed out on growing up. I have recommended this to many of my homeschool friends.
This is a really fun way for kids, well anyone really, to read about math. I read it for a college course in a secondary education program (although I am English and not Math), and am currently working with a group to create an interdisciplinary unit centered around this book. It's challenging, but we're finding some really fun ways to create 3-week units for 3 different subjects: English, Math, and Art.
This book is a wonderfully funny way to talk about math. My 8 yr. old daughter is thrilled about reading the next "night/chapter", even though some of the math concepts are way beyond her level. It's an excellent book!!
The Number Devil was a great book. I had to read it for a project in math and found it more than enjoyable. Some of the math concepts were a little confusing but managable.
I am in seventh grade. We read this book to learn and have fun. I had never imagined what had awaited me !(vroom)
I'm a 6th grader and i just love to read this book for fun!! it was hidden away for a few years and when i remembered it again, itook it off the shelf again.
This is the summer reading book for our incoming 8th graders. Writing Across the Curriculum is a key component of our PreK-12 school. The 8th grade teachers have developed activities to go with many of the chapters. We have the students work in groups throughout the school year on these and then each student is required to write an essay or creative story to explain how they found (or tried to find) the answers to the questions posed. We start off with the students finding out about the pattern with the multiplication of ones (1x1, 11x11, 111x111, etc.), then go on to exploring the "Prima Donnas", terminating and repeating decimals, square numbers, geometric patterns in Pascal's Triangle along with its relationship with the binomial expansion and also permutations and combinations, and summation. As our "last chapter", the students pretend they are a Number Devil and research a topic to explain to someone as their first assignment as a number devil. It's a great book because it is easy to read and gets kids to think about mathematical ideas. It can be read by people of many ages. I read it to my children as 3rd and 5th graders when we were driving to Nova Scotia. My son then read it again for a project when he was in 5th grade. I would love to have a sequel!
I really loved this book, and as a future math educator, I found it to be very useful for something I could one day use in the classroom. The only thing that bothers me is that they use fake and "quirky" names for the mathematical concepts. It was ok for me, because I know what they are actually called; however, if a student was reading the book for themselves, they may have issues translating those ideas when they see them in class. I really wish that the book would have just put the actual names of the concepts, and not tried to be as cute. I did really like the way the concepts were presented and thought it was a great way to get kids interested in math.
This is without doubt one of the best mathematical fictions I've read. I frequently re-read it just for fun, and every so often, I'll get into a conversation about it with one of my "geeky" friends.
It presents mathematical concepts in an easy-to-understand but NOT dumbed-down way for children who aren't yet able to read math textbooks for fun (yes, there are people like that!) It does not use the common names for most of the concepts it introduces, but it does have a glossary/index/translation in the back.
Examples of differences in terminology:
Bonnacci numbers -- Fibonacci series
Hopping numbers -- exponents
Rutabaga -- root
Owler -- Euler
Horrors -- Gauss
As a future middle school math teacher, I am thrilled to have discovered Number Devil. I can't wait to create lessons that incorporate this book. The list of historical figures in the back of the book is a great reference and a nice segue into a unit plan for integrating writing and reading into a math classroom.
My fourth grade daughter and I alternated reading a chapter aloud every day. It was a funny and interesting book that made my math phobic daughter think of math in a friendlier way. I don't think she LOVES math...yet, but at least she doesn't hate it so badly. I'm sure we'll read it again.
I do have a question about the concepts covered in the book. Are the concepts such as Fibonacci numbers and Pascal's triangle merely mathematical "recreations" or are they actually useful for something?
Maria, that is a great question. I think I see exactly what you mean. There are parts of mathematics that clearly have an application (like computing your income taxes or landing a space ship on the moon) and other parts of math that seem more like puzzles serving no other purpose than the enjoyment of the mathematicians playing with them.
There are in fact many applications for Pascal's Triangle, because the numbers in its rows are the answers to two very practical types of questions. Before I get to that, however, I need to set up a sort of strange terminology. Let's say that the top of the triangle ("1") is the 0th row and that the next row ("1 1") is the 1st row and the next row ("1 2 1") is thesecond row and so on...since that will make it easier for me to explain. And similarly, let's call the 1 at the start of each row the "zeroeth" number in that row and the number after it the first number. So, according to this strange notation, the first number in the second row is 2...right? Okay. Now, I can show you an application of the triangle.
Suppose I have four socks (sock 1, sock 2, sock 3 and sock 4) in a drawer. If I reach in and without looking pick two socks, how many possibilities are there for which two socks I get? As you can see by counting all of the possibilities, there are SIX possible pairs because I could get 1&2, 1&3, 1&4, 2&3, 2&4, or 3&4...there are six of them. This is connected to the fact that the number in the second spot in the fourth row of the triangle is six!
In general, we can say that the number of ways of picking k objects out of a set of n objects is given by the number in the kth spot of the nth row of the triangle. This becomes more impressive when the numbers are larger and you cannot easily count all of the possibilities.
Suppose I have invited 20 guests to a party but only have enough seating for 10. How many different possibilities are there for the list of 10 people who get to sit? It would take too long to try to list all possible choices of 10 names of the 20 guests. Instead, we should use the triangle. The number of possible sets of 10 selected out of a larger set of 20 would be the number in the 10th spot in the 20th row of Pascal's Triangle: 184756.
This procedure is frequently needed by people computing probabilities. For instance: the odds of winning the lottery. If your lottery asks you to pick 5 numbers from 1 to 60, then your odds of getting the same 5 numbers as the ones that pop up on TV are 1 in 5461512 (that is, less than one in five million) because that is the number in the 5th spot of the 60th row of the table. It is because of this that Pascal's triangle can be described as having a similar significance as the famous "bell curve" that you have probably heard about. (See this other website for a more careful explanation of this idea.)
Another type of question that is answered by Pascal's Triangle has to do with algebra. Most algebra students know that (x+y)2=x2+2xy+y2. Note that the coefficients in this expansion are the same as the numbers in the second row of the triangle! More generally, you can read off the coefficients of (x+y)n from the nth row of the table. The fourth row is "1 4 6 4 1" which means that (x+y)4=x^4 + 4 x3y+6x2y2+4xy3+y4.
The Fibonacci series is not quite as broadly useful, but it also has connections to the real world. In particular, biology seems to have "discovered" the usefulness of Fibonacci numbers long before we did, which explains why we can find them so often when counting petals or seeds on plants. Specifically, we can show that the Fibonacci numbers arise when we try to figure out how to optimally pack things around in a circle, as plants want to do with their own parts. Furthermore, the Fibonacci numbers are related to the aesthetics espoused by the ancient Greeks (the Golden Ratio) and the ratios of male to female bees in bee colonies! For more detailed explanations of these ideas, please see here and here.
That's my answer to your question. But, before you leave, let me offer one more bit of philosophy. Again, I do understand that some bits of mathematics appear to be much more useful than other bits. Some things that mathematicians study and play with may appear to be totally useless. I would first like to argue that there is a value in exploring the "mathematical landscape" even if no obvious application is apparent...just like we consider it to be valuable for geographic explorers to look around at places that no person has visited just to see if there is anything useful or interesting there. But, most amazingly, I'd like to add that it has happened quite often that mathematicians were studying something that appeared to be completely useless, and it later turned out to have been incredibly useful in ways that nobody had imagined. D'Alembert studied the mathematics of how violin strings vibrate, which may seem unimportant since people were making perfectly good violins without him...but it turns out to have been an important step in our later discovery of radio waves. Imaginary numbers are frequently mentioned as being "unrealistic", but we use them to describe the light waves that we send through fiber optic cables in the latest advances in communications. Mathematicians in the 19th century studied "non-Euclidean geometry", which I'm sure was considered the height of ridiculousness...but since Einstein's work on general relativity we know that the space we live in is governed by exactly that sort of geometry! I could go on and on, but the point is that I would not belittle any area of math as "just recreational" because almost all of it seems to find surprising applications somehow and sometime. (I wrote a story about how weird that is, in fact. See here.)
Thanks for writing...I hope my answer was not too long. Please do feel free to ask for additional explanations if I have not been clear.
I just LOVED this book (I learned factorials and the Fibonacci sequence on it, for instance).
One thing upsets me, however: in the Italian translation the title reads as "The Number magician" - actually, until I found the original title on this page, I've been wondering why the magician looked so much like a devil in the pictures. Why this censorship? Did they think that a devil would scare Italian children, or that it would enrage the Catholic Church?
John C. Konrath|
Mr. Enzenberger does a marvelous job of making seemingly difficult number theory concepts fun and easy to understand. This book could be equally enjoyable to both adults and children and the color illustrations are helpful and artistically appealing.
My largest concern, like many other critics, is the author's renaming of terms. I can just imagine the poor kid's grief when he/she raises his/her hand and asks the teacher the "rutabaga" of two. Furthermore, two questions that came to mind were: Why a devil? & Why is poor Mr. Bockel portrayed in this manner?
I picked this up first of all to practice my German, but was instantly hooked. It's charming and witty. I've been giving copies (in appropriate languages) to children ever since.